Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 20 + \left(28 a + 37\right)\cdot 41 + \left(21 a + 10\right)\cdot 41^{2} + \left(39 a + 15\right)\cdot 41^{3} + \left(39 a + 12\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 3 + \left(23 a + 4\right)\cdot 41 + \left(21 a + 21\right)\cdot 41^{2} + \left(12 a + 24\right)\cdot 41^{3} + \left(4 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 16 + \left(17 a + 15\right)\cdot 41 + \left(19 a + 21\right)\cdot 41^{2} + \left(28 a + 40\right)\cdot 41^{3} + \left(36 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 12 + \left(4 a + 34\right)\cdot 41 + \left(38 a + 37\right)\cdot 41^{2} + \left(2 a + 20\right)\cdot 41^{3} + \left(6 a + 35\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a + 30 + \left(36 a + 40\right)\cdot 41 + \left(2 a + 24\right)\cdot 41^{2} + \left(38 a + 32\right)\cdot 41^{3} + \left(34 a + 9\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 3 + \left(12 a + 32\right)\cdot 41 + \left(19 a + 6\right)\cdot 41^{2} + \left(a + 30\right)\cdot 41^{3} + \left(a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4,6)$ |
| $(1,5,3)$ |
| $(1,6,5,4,3,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,5,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,3,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,3)(2,4,6)$ | $-1$ |
| $3$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
| $3$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
